Cyclicity of nonvanishing functions in the polydisk and in the ball

A special version of the corona theorem in several variables, valid when all but one of the data functions are smooth, is used to generalize, to the polydisk and to the ball, the results obtained by El Fallah, Kellay, and Seip about the cyclicity of nonvanishing bounded holomorphic functions in sufficiently large Banach spaces of analytic functions determined either by weighted sums of powers of Taylor coefficients or by radially weighted integrals of powers of the modulus of the function.